THE SAMPLING PROCESS 23 
that the pulse resulting from a finite sampling time is flat-topped. It is 
recognized that this characterization is not precise since the actual pulse 
would have a top which follows the actual function being sampled. The 
effects produced by non-flat-topped pulses are of second order in so far 
as low-pass system are concerned and are of little consequence in prac- 
tical problems. 
While the details of data reconstruction are discussed in the next 
chapter, certain preliminary considerations should be reviewed as to their 
effect on the assumption of impulse sampling. In practice the more 
common situation is that the pulse sequence is applied to some type of 
“data hold.” The function of such an element is to reconstruct, in so far 
as possible, the original continuous function from which the pulse 
sequence was derived. Typical of such a device is the simple clamp 
circuit which accepts a short pulse and “‘stretches”’ it to a duration equal 
to the sampling interval. The effect of such a clamp can be reproduced 
mathematically using the impulse approximation by assuming that the 
clamp circuit responds to the area of the impulse rather than the magni- 
tude of the pulse. This simple mathematical strategem introduces no 
error and makes possible the simplifications of the impulse approximation 
with little or no loss of accuracy. For this reason, the impulse approxi- 
mation is almost always used in the analysis and synthesis of sampled- 
data feedback control systems. 
2.4 Laplace Transform of the Impulse Sequence 
As in the case of the continuous system, the Laplace transform finds 
considerable utility in sampled-data systems. As a result it is desirable 
to consider the Laplace transform of an impulse-modulated function 
f*(t). If it is assumed that the function f(¢) from which f*(é) is derived 
is Laplace transformable, and it is zero for negative time then f*(t) is 
iO = ) fr) — n7) (2.21) 
n=0 
where f(n7) is the value of f(t) at the sampling instant nT. Thus a 
typical term of the sequence is an impulse at time nZ’ whose area is 
f(nT). The Laplace transform of this typical term is 
L[f(nT)6¢ — nT)] = f(nT)e?s (2.22) 
The Laplace transform of the impulse sequence F'*(s) is thus 
+o 
F*(s) = » f(nT)e*?s (2.23) 
n=0 
